St. Louis Rams beat the Oakland Raiders in week 13 (of 2014 season) by a score of 52 to 0, and went on to beat Washington Redskins the following week 24 - 0.
Are the Rams the first team to win back to back games without allowing their opponents to score any points? If not, what other teams accomplished this feat; and what is the longest streak of holding your opponents to no points in the NFL?

2 Answers
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According to this Wikipedia list of NFL records by team, the "Most Consecutive Shutout Games Won or Tied" was 13 by the Akron Pros over the 1920 and 1921 seasons. They finished the 1920 season with 5 straight shutouts (the last 2 were ties), and they started the 1921 season with 8 straight shutouts (the last one was a tie).

The only NFL teams (since 1940) to have three consecutive shutouts were the 1970 Cardinals and the Steelers, who did it during Games 7-8-9 of the 1976 season and closed out that season with a pair (Games 13-14) and started 1977 with one (Game 1).

Yep, the era is important. In the early days, shutouts were very common, as were 0-0 ties. Lower skill level combined with no passing game meant for a very low scoring game (which apparently people were more apt to watch back then in the US than they are to watch soccer 0-0 ties now?)
– JoeDec 8 '14 at 15:18

In modern NFL football, this would be the Miami Dolphins, who shutout the Baltimore Colts 4 consecutive times, from January 2, 1972 (AFC Championship) to November 11, 1973.
The cumulative score of all four games was 104-0.